question_answer

ABCD is a cyclic quadrilateral and AD is a diameter. If \[\angle DAC=55{}^\circ ,\]then the value of \[\angle ABC\]is

A) \[55{}^\circ \]

B) \[35{}^\circ \]

C) \[145{}^\circ \]

D) \[125{}^\circ \]

Correct Answer: C

Solution :

[c] In \[\Delta ACD\] \[\angle DAC=55{}^\circ \]                                [given] \[\angle ACD=90{}^\circ =\] angle in a semicircle \[\angle ADC=180{}^\circ -90{}^\circ -55{}^\circ \] \[=180{}^\circ -145{}^\circ =35{}^\circ \] Now, in a cyclic quadrilateral Sum of opposite angles \[=\text{ }180{}^\circ \] \[\angle ABC+\angle ADC=180{}^\circ \] \[\angle ABC=180{}^\circ -\angle ADC\] \[=180{}^\circ -35{}^\circ \] \[=145{}^\circ \]